Multidimensional Residues and Applications

One of the problems in the theory of multidimensional residues is the problem of studying and computing integrals of the form $$\int_\gamma {\omega ,} $$ (1) where ω is a closed differential form of degree p on a complex analytic manifold X with a singularity on an analytic set S⊂ X, and where γ is a compact p-dimensional cycle in X\S. A special case of this problem is computing the integral (1) when ω is a holomorphic (meromorphic) form of degree p = n = dimc X; in local coordinates the form can be written as ω = f(z) dz = f(z 1, ...,z n )dz 1 ∧ ... ∧ dz n , where f is a holomorphic (meromorphic) function. According to the Stokes formula, the integral (1) depends only on the homology class1 [γ] ∈ H p (X\S) and the De Rham cohomology class [ω] ∈ H P (X\S). Thus in integral (1) the cycle γ can be replaced by a cycle γ1 homologous to it (γ1 ~γ) in X\S and the form ω can be replaced by a cohomologous form ω1(ω1 ~ ω) which may perhaps be simpler; for example, it could have poles of first order on S (see § 1, Subsection 4). If γj is a basis for the p-dimensional homology of the manifold X\S, then by Stokes formula for any compact cycle γ ∈ Z p (X\S) the integral (1) is equal to $$\int_\gamma \omega = \sum\limits_j {k_j \int_{y_j } {\omega ,} } $$ (2) where the k j are the coefficients of the cycle γ as a combination of the basis elements γ j , γ ~ Σ j k j γ j. Formula (2) shows that the problem of computing integral (1) can be reduced t 1) studying the homology group H p (X\S) (finding its dimension and a basis); 2) determining the coefficients of the cycle γ with respect to a basis; 3) computing the integrals over the cycles in the basis.